The nineteen decades a slow coupling buys

Where every model runs out at once found the Planck mass by asking where a particle’s quantum size and its gravitational size coincide: about twenty-two micrograms, or $1.22 \times 10^{19}$ GeV in the units particle physics uses. Every elementary particle is far below it. The proton, at 0.94 GeV, is nineteen orders of magnitude lighter, and that gap is why gravity is so feeble in the world of atoms. Two protons attract each other gravitationally with a strength, measured in the natural units in which the electric coupling is 1/137, of $(m_p/M_P)^2 = 5.9 \times 10^{-39}$.

The scale that may not be where it looks took that gap as a problem and examined one proposed solution, that gravity leaks into extra dimensions so that its true scale is lower than it appears. There is a second reading of the same gap, older and entirely established, in which nothing about gravity needs explaining at all. On this reading the Planck mass is simply where gravity is; it is the proton that is anomalous, and the reason it is so light is a logarithm.

A coupling that depends on the distance #

The strong interaction, which binds quarks into protons and neutrons, has a coupling constant that is not constant. Probed at short distances, by collisions at high energy, it is weak: about 0.118 at the energy of the Z boson, 91 GeV, and smaller at higher energies. Probed at long distances it is strong, so strong that quarks cannot be separated. The figure plots the inverse of the coupling against the energy of the probe, and on a logarithmic energy axis the plot is nearly a straight line. At the lowest order of approximation it is exactly one, in each range between quark masses:

$$\frac{1}{\alpha_s(\mu)} = \frac{1}{\alpha_s(\mu_0)} + \frac{\beta_0}{2\pi}\,\ln\frac{\mu}{\mu_0}, \qquad \beta_0 = 11 - \tfrac{2}{3} n_f,$$

where $n_f$ is the number of quark types light enough to matter at energy $\mu$. Each factor of ten in energy adds a fixed amount to $1/\alpha_s$, between 2.1 and 2.9 depending on how many quarks are active, and the slope changes slightly at each quark’s mass. The 11 in $\beta_0$ comes from the gluons, which carry the strong charge themselves and so make it grow with distance; the $-\tfrac23 n_f$ comes from the quarks, which screen it as electrons screen an electric charge. The gluons win as long as there are fewer than seventeen kinds of quark, and there are six.

That the coupling grows weaker at short distance, asymptotic freedom, was found by David Gross, Frank Wilczek and David Politzer in 1973 and earned them the 2004 Nobel Prize. Its measured form, from collisions spanning a few GeV to a few TeV, follows the running to within the experimental uncertainties.

Electricity runs too, and in the opposite direction, which is the clearest way to see why the sign matters. The vacuum around an electron is full of fleeting electron–positron pairs, and they polarise as the molecules of a dielectric do, partly cancelling the electron’s charge as seen from far away. The long-range force that does not reach found the same screening in a plasma, where mobile charges gather round a test charge and hide it beyond a Debye length. In the vacuum the screening is weak and logarithmic: the electric coupling, 1/137 at long distances, has grown to about 1/128 at the Z mass. Photons carry no charge, so nothing opposes the screening, and electricity’s coupling strengthens at short distance and weakens at long. It never becomes strong at low energy, and so electromagnetism makes no mass scale of its own. The gluons are different because they carry the strong charge — the colour that the label three identical quarks forced introduced to rescue the quark model — and a cloud of charged gluons round a quark spreads its charge out rather than hiding it. Seen from further away, the quark looks more strongly charged, not less. That reversal, from screening to its opposite, is the whole difference between a force that leaves its particles light and free and one that builds a mass out of nothing and confines what it acts on.

Run upward, the line reaches the Planck mass at $1/\alpha_s = 52.5$, a coupling of about 1/50: small but not remarkable, a number of the same order as the electric coupling’s 1/137. Run downward, $1/\alpha_s$ falls by the same step per decade and reaches zero at a definite energy, 143 MeV at this order of the calculation. There the coupling formally becomes infinite and the approximation that produced the straight line collapses. That energy, written $\Lambda$, is where the strong interaction becomes strong.

The scale nobody put in #

The equation for the running contains no mass. A theory of massless quarks and gluons has no length or energy in it at all, and yet the running coupling defines one: the energy at which the coupling, starting from whatever value it has anywhere, reaches order one. Solving the straight line for that energy gives

$$\Lambda = \mu\,\exp\!\left(-\frac{2\pi}{\beta_0\,\alpha_s(\mu)}\right),$$

and the right-hand side is the same whatever reference energy $\mu$ is chosen, which is what makes $\Lambda$ a property of the theory rather than of the measurement. Sidney Coleman and Erick Weinberg named the phenomenon dimensional transmutation: a pure number, the coupling, is traded for a scale.

The scale is the proton’s. A proton is two up quarks and a down quark, but their masses — the part that does come from the Higgs field — are 2.2, 2.2 and 4.7 MeV, nine in all, about one per cent of the proton’s 938. Calculations of the proton’s mass on a lattice of spacetime points, which solve the strong interaction without approximation, reproduce the proton’s mass to about two per cent and find that it would be only slightly smaller if the quarks were massless. The proton’s mass is the energy of three quarks confined in a region about a femtometre across and of the gluon field that confines them, and both are set by $\Lambda$, because $\Lambda$ is the only energy the strong interaction has.

The second figure sets the numbers side by side. The ratio of the proton’s mass to the one-loop $\Lambda$ is 6.6; with the more accurate running of higher orders, $\Lambda$ for three light quarks comes out near 340 MeV and the ratio near 2.8. Neither number is fundamental — $\Lambda$ depends on a convention for how it is defined — but the point survives every convention: the proton is a few times $\Lambda$, and $\Lambda$ is a consequence of the coupling and its running.

The electron is the contrast. It feels no strong force, so no running coupling makes its mass. It weighs 0.511 MeV because its coupling to the Higgs field is about $3 \times 10^{-6}$, and that number is simply put in. The Standard Model has no explanation for it, and why an atom is the size it is took the electron’s mass as given for exactly that reason.

Why fifty is enough #

The third figure asks how the proton’s mass depends on the one number that sets it. Fix the coupling at the Planck mass and run it down. The scale where it runs off falls exponentially as the starting coupling weakens: each unit added to $1/\alpha_s$ at the top divides $\Lambda$ by about two. The value 52.5 that the measurements imply gives 143 MeV; a value of 51.5 would give 286, and 53.5 would give 71. Going from $1/\alpha_s \approx 30$ to $\approx 75$ spans twenty decades of $\Lambda$.

This is what makes the hierarchy unremarkable. To get a proton nineteen orders of magnitude below the Planck mass, the theory needs a coupling of about 1/50 at the Planck mass, and 1/50 is an ordinary number. Nothing has to be adjusted to nineteen decimal places. A two per cent change in the coupling doubles or halves the proton’s mass, which is a strong dependence, but it is the dependence of an exponential, not of a cancellation. Large ratios come cheaply out of exponentials: the same arithmetic makes radioactive half-lives range over fifty orders of magnitude from modest changes in barrier height, as a wall that a factor of two makes impassable found for alpha decay.

The contrast with the Higgs boson is the reason particle physicists worry about one and not the other. The Higgs boson’s mass is not protected in this way. Quantum corrections to it grow with the square of the highest energy the theory describes, so if the Standard Model held up to the Planck mass, the measured 125 GeV would have to emerge from a cancellation between quantities of order $10^{19}$ GeV squared to one part in about $10^{33}$. The estimate that misses by a hundred and twenty found a cancellation of the same kind, worse by ninety orders of magnitude, in the energy of the vacuum. That is the hierarchy problem, and the strong interaction is the existence proof that a theory can produce a large hierarchy without having one. Many of the proposed solutions to the Higgs problem try to make the Higgs mass arise from a running coupling in exactly the way the proton’s does.

The weakness of gravity, relocated #

The fourth figure follows the consequence. The strength of gravity between two protons, in the same units as the electric 1/137, is $Gm_p^2/\hbar c$, which is the square of the proton’s mass measured in Planck masses. Since the proton’s mass is set by $\Lambda$, so is gravity’s coupling between protons, and it inherits the exponential: each unit of $1/\alpha_s$ at the Planck mass changes it by a factor of four. At the measured value it is $5.9 \times 10^{-39}$.

Read this way, the famous weakness of gravity is not a property of gravity. Newton’s constant has units, so it has no size of its own; it simply fixes where the Planck mass lies. Two particles of the Planck mass attract gravitationally with a coupling of one, as strong as anything. Gravity looks weak in atoms because atoms are made of protons, and protons are made light by the slow running of a different force. The ratio $10^{36}$ between electric and gravitational attraction, which the fall that does not depend on what is falling and every textbook quote as a measure of gravity’s feebleness, is a measure of how slowly the strong coupling runs.

The same exponential sits under the size of everything made of atoms. The size at which a body becomes round and the mass of stars depend on how weak gravity is compared with electrical forces in matter, and so, through the proton’s mass, on a coupling of 1/52.5 at the Planck mass. A universe whose strong coupling started at 1/40 would have protons six thousand times heavier and gravity between them forty million times stronger, and its stars would be correspondingly smaller and shorter-lived. Nothing in the figures says such a universe is possible or impossible; they say only how steep the dependence is.

A particle the proton never contains #

The last figure shows a strange consequence of the running. The proton contains no top quarks; the top is 184 times heavier than the proton and lives for less than a trillionth of a trillionth of a second. Yet holding the coupling at the Planck mass fixed, the proton’s mass depends on the top’s. Above the top’s mass six kinds of quark screen the strong charge and the coupling runs at one rate; below it, five, and a slightly faster rate. Move the top’s mass and the switch between rates moves, so the coupling arrives at low energy slightly different. The dependence works out, at one loop, to a power law with the curious exponent $2/27$, fixed entirely by the counting of quark flavours on either side of each threshold, and it gives about five per cent in the proton’s mass for a doubling of the top’s.

This is the same structure as the main argument, in miniature: a heavy particle can influence low-energy physics without appearing in it, but only through the logarithm of its mass, so the influence is weak. It is one reason the strong scale could be measured accurately before the top quark was found, and one reason that precise measurements of $\alpha_s$ at low energy constrain what heavy particles might exist above the energies colliders reach: each would change the running above its own mass.

How far the straight line can be trusted #

The running drawn in these figures is the first term of an expansion. Higher terms bend the lines slightly, change $\Lambda$ by a factor of two or so, and depend on conventions for how the coupling is defined; none of that alters the exponential dependence, which is the whole argument. Near $\Lambda$ itself the expansion fails completely — the line reaching zero is the approximation announcing that it has nothing more to say — and the proton’s mass has to be computed by other means. The extrapolation in the opposite direction, from the 100 GeV at which $\alpha_s$ is measured up to the Planck mass, assumes that no particles beyond the known ones change the running in between. That is not known. Heavy particles that carry the strong charge, such as those predicted by supersymmetry, would change the slope above their mass, and the value of $1/\alpha_s$ at the Planck mass would then differ from 52.5. The conclusion that the proton’s mass needs only a coupling of order 1/50 survives any such change; the specific number does not.

The extrapolation also assumes that the Planck mass is where the coupling should be specified, and nothing guarantees that the coupling’s value there is set by anything at all. If the three forces of the Standard Model are unified at some higher energy, their couplings meet there and the strong coupling at the Planck mass follows from the unified one, an attractive idea that the running of the three measured couplings supports approximately and does not confirm.

What a running line on a page leaves unsaid #

None of the figures shows a proton. They show a coupling and a scale, and the step from the scale to the proton — the actual binding of three quarks into a particle that weighs 938 MeV — happens in the region where the straight line has failed. That step has been taken numerically, on lattices, and it works; there is still no analytic understanding of confinement, and proving that the theory of quarks and gluons produces a mass gap at all is one of the Clay Mathematics Institute’s millennium problems, unsolved. The figures also say nothing about why the coupling at the Planck mass has the value it has. They relocate the question from “why is the proton so light?” to “why is $1/\alpha_s$ about fifty?”, which is a much less demanding question but not an answer.

Still open: whether the Higgs mass is made the same way #

The proton shows that a running coupling can make a light particle out of a theory with no small numbers. Whether the Higgs boson’s mass, the other scale far below the Planck mass, has a similar origin is the central open question about the electroweak scale. Theories in which the Higgs is a composite, bound by a new strong force whose coupling runs to strength near a TeV, would make it so, and the Large Hadron Collider has looked for the new particles such theories predict without finding them up to energies of a few TeV. Other proposals keep the Higgs elementary and protect its mass by a symmetry. Which, if either, is right has not been settled, and the hierarchy between the Higgs mass and the Planck mass, unlike the one the proton inhabits, remains unexplained.

The habit worth carrying away is to ask how a large ratio depends on its inputs. The proton is $10^{19}$ times lighter than the Planck mass, and gravity between protons $10^{36}$ times weaker than electricity, because the strong coupling — about 1/52.5 at the Planck mass, growing by a fixed step per decade of lower energy — takes nineteen decades to become strong: an exponential in an ordinary number, $\Lambda = \mu\,e^{-2\pi/\beta_0\alpha_s}$, where a cancellation would have needed nineteen decimal places. A huge ratio from an exponential is cheap; a huge ratio from a power is a puzzle.